Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661-674] characterized invertible reverse-ordering transforms on the space of lower semi-continuous extended real-valued convex functions as affine deformations of the ordinary Legendre transform. In this work, we first prove that all those generalized Legendre transforms on functions correspond to the ordinary Legendre transform on dually corresponding affine-deformed functions: In short, generalized convex conjugates are ordinary convex conjugates of dually affine-deformed functions. Second, we explain how these generalized Legendre transforms can be derived from the dual Hessian structures of information geometry.

翻译：Artstein-Avidan与Milman [Annals of mathematics (2009), (169):661-674] 将下半连续扩展实值凸函数空间上可逆反序变换的特征刻画为经典勒让德变换的仿射形变。本文首先证明：函数上的所有这类广义勒让德变换，均对应于对偶相应仿射形变函数上的经典勒让德变换——简言之，广义凸共轭即对偶仿射形变函数的经典凸共轭。其次，我们阐释这些广义勒让德变换如何从信息几何的对偶Hessian结构中导出。